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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク perpendicular axis theorem ： ウィキペディア英語版 perpendicular axis theorem In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane. The axes must all pass through a single point in the plane. Define perpendicular axes $x\backslash ,$, $y\backslash ,$, and $z\backslash ,$ (which meet at origin $O\backslash ,$) so that the body lies in the $xy\backslash ,$ plane, and the $z\backslash ,$ axis is perpendicular to the plane of the body. Let ''I''''x'', ''I''''y'' and ''I''''z'' be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that :$I\_z\; =\; I\_x\; +\; I\_y\backslash ,$ This rule can be applied with the parallel axis theorem and the stretch rule to find moments of inertia for a variety of shapes. If a planar object (or prism, by the stretch rule) has rotational symmetry such that $I\_x\backslash ,$ and $I\_y\backslash ,$ are equal, then the perpendicular axes theorem provides the useful relationship: :$I\_z\; =\; 2I\_x\; =\; 2I\_y\backslash ,$ == Derivation == Working in Cartesian co-ordinates, the moment of inertia of the planar body about the $z\backslash ,$ axis is given by: :$I\_\; =\; \backslash int\; \backslash left(x^2\; +\; y^2\backslash right)\backslash ,\; dm\; =\; \backslash int\; x^2\backslash ,dm\; +\; \backslash int\; y^2\backslash ,dm\; =\; I\_\; +\; I\_$ On the plane, $z=0\backslash ,$, so these two terms are the moments of inertia about the $x\backslash ,$ and $y\backslash ,$ axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly. Note that $\backslash int\; x^2\backslash ,dm\; =\; I\_\; \backslash ne\; I\_$ because in $\backslash int\; r^2\backslash ,dm$, r measures the distance from the ''axis of rotation'', so for a y-axis rotation, deviation distance from the axis of rotation of a point is equal to its x co-ordinate. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「perpendicular axis theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.